Computational Calculations: a linkage of statistical thermodynamics and experiments
Introduction
Physical chemists usually use thermodynamic parameters to describe systems of interest. Typically temperature, pressure, and volume are used since these are the macroscopic variables that could be directly measured in the laboratories. Furthermore, only two out of the three variables mentioned above are needed to measure and thereby the studied systems could be described completely (1). Thermodynamics is indeed a very powerful tool for studying Chemistry under the macroscopic point of view. Not only provides direct quantities that could be measured from experiments, Thermodynamics could also show mathematical relationships between these quantities thereby we could use these relationship to understand the characteristics or properties of the systems under investigation. However, in the study of Thermodynamics, we do not consider the magnitude of the parameters and we also do not need to care about the fundamental particles which are the basic components of the system (2). From a molecular point of view, we need to describe the studied system in more details and we also need to consider the existence of atoms and molecules. Another area called statistical thermodynamics therefore is needed to bring into the study of Chemistry. Statistical thermodynamics is a discipline that has to deal with the calculation of bulky properties from a molecular point of view (2). In statistical thermodynamics, the concepts of ensembles, partition functions, Boltzmann statistics, and etc. are used to describe systems. We theoretical could calculate whatever system we would like to study. However, it is not practical to do so most of the time. First, assumptions still are needed to put into equations when the calculations are carried out in the study of statistical thermodynamics. Second, there are still some unsolved conceptual problem in statistical thermodynamics such as one is dealing with non-equilibrium systems although there have been so much effort were put into such kinds of system since mid-20th-century. Third, most importantly, it is impossible to carry out calculation on some complicated systems such as biological systems. Fortunately, since the World War II computers were introduced to scientist and thereby methods of calculations using statistical thermodynamics were started to developed. Computers are not necessary smarted than human but they are very fast of calculating equations and computers are good at not making mistakes. With the increasing computing power, scientists could study biological systems in more detail than ever before. For example, computational calculations of solving Newton's second law, Monte Carlo method, Potential Mean force calculations, and etc. are able to carry out to investigate complex system such protein folding and ion channel conformational change. Not being the calculations are impractical to perform by hand, it is also important to point out that it is typically a huge number of sample calculations we need to do and then take the average of these calculations, and out of such average we could minimize the statistical error or fluctuation in which we could be certain of our calculation. Such calculations are extremely long and are almost impossible to perform without computers. With the statistical thermodynamics in hand as a theoretical background and the utilization of advance computer developments, scientists therefore can study chemistry in the molecular level quantitatively. This article's focus is to introduce the methods of computational calculations and analysis of the results based of statistical thermodynamics calculations. The examples include molecular dynamics simulations, potential mean force calculations, and their role of being the linkage of experiments and statistical thermodynamics. More precisely, a project on computational investigation on a biological system will be given in this article as an example to show how theoretical calculations was done in the background of statistical thermodynamics and how the results are compared to the experimental works.
Background
In order to calculate the macroscopic properties from the microscopic point of view, some important mathematical concepts are needed to be introduced. First mathematics concept is the probability distribution. It is important to understand the random variables and the distribution function which are needed to be introduced in the calculations of statistical thermodynamics. The average, or the mean value, is a typical value obtained from calculation in the study of statistical thermodynamics. Moreover, the maximum term method is another concept will be applied when dealing with statistical thermodynamics. Since large numbers are usually involved in statistical thermodynamics calculation, it is appropriated to use the largest term, typically much larger comparing to other terms, as the peak out of all the probability values. The third mathematical concept is the Stirling's approximation because in statistical thermodynamics we usually have to deal with factorials of large numbers. The Stirling’s approximation (2) is the following, lnN! = NlnN-N, where N is a very large number (#1) Binomial and multinomial distributions, which are needed to use where the ensemble idea is used in statistical thermodynamics, could be used to determine how many states we encounter of many distinguishable system in a group under investigation. Especially the binomial distribution for large number is needed for statistical thermodynamics calculations because it is very important that one cal find the maximum value of functions. Finally, the method of Lagrange multipliers comes handy when we have to maximize variables since most of the times we need to find the maximum of certain functions or to set the derivative of certain function to be zero and solve for needed variables. With all the mathematical concepts above on mind, now the base of statistical thermodynamics calculation is set and we are ready to introduce some important concepts of statistical thermodynamics. One of the important concepts that statistical thermodynamics introduced is ensemble average (2). To view the macroscopic properties such temperature, pressure, and volume from the molecular point of view, the idea of ensemble is introduced in the late 19th century. To calculate the bulky properties from the microscopic quantities, scientists introduced the concept of ensemble, in which the studied system is composed of a large numbers of identical subsystems. And these subsystems are then treated as equal parts of the studies systems. What we have to do is just to calculate the properties of one subsystem out of all these subsystems to understand the whole studied system. Popular ensembles (2) used in statistical thermodynamics are microcanonical ensemble, canonical ensemble, grandcanonical ensemble, and isothermal-isobaric. Microcanonical ensemble (NVE) is system consist constant number of particle (N) with fixed energy (E) and a fixed volume (V). Under the constrain of constant number of particle and fixed energy and fixed volume, the partition function with correlated thermodynamics parameter could be able to identified using the concept of ensemble average and mathematic concepts mentioned above. Similarly to microcanonical ensemble, grandcanonical (µVT) ensemble, canonical (NVT) ensemble, isothermal-isobaric (NPT) ensemble could be also used to calculate macroscopic properties, where µ is the chemical potential, T is the temperature, and P is the pressure (2), representatively. The most important thing we could be able to get out of ensemble average is the partition function of a system because using the partition function one can derive the thermodynamic function and thereby calculate the thermodynamics properties. Therefore the bulky properties could be calculated from the point of view in the molecular level. With the concepts of statistical thermodynamics and mathematics introduced, we have to realized that the ensemble average calculations is easy but time consuming when we have to deal with large and complicated system such as biological system. It is not practical to do such calculations over and over again by hand and it is almost impossible to do so most of the case. There is where computational calculation comes into place. The calculations performed on computers involves molecular dynamics (3) (MD) simulation on reaction dynamics, quantum mechanic calculations on electronic structural information, Monte Carlo calculation on random walk simulations, potential mean force calculation for estimating reaction free energies, and etc. In this article, MD simulation and PMF calculations will be described as the linkage of experiments and statistical thermodynamics. Many experimental works had been carried out to investigate the mechanism of certain biological system such as anesthetics binding to lipid, however, due to the limitation of experiments, some perspectives in the molecular level are difficult to determine. For example, the binding mode of reaction is usually hard to reveal since the binding time is relative short and it is hard to detect by instruments. To access molecular detail, molecular dynamics simulations can be utilized. In molecular dynamics, the atoms interact with each other respect to time. The calculations of the motion of these atoms are solved using the Newton’s second law of motion: F(x) = m ×a = -∇U(x) (#2) where in eqution (#2), F(x) is function of the force act on particle respect to the position, x , m is the mass of particle, a is the acceleration of the particle and ∇U(x) is the potential energy of the particle. These coupled differential equations are solved using numerical integration on the computers. From the initial position and velocity, MD simulations are used to calculate or predict the future positions, velocities, and potential energies. The time revolution of position and velocity is called the trajectory. Typically, the simulations are using either one of the following ensembles: NVE, NVT, or NPT ensemble, whereas N, V, P, T,and E stands for constant numbers of atoms, volume, pressure, temperature, and energy, respectively. These ensembles are any different from the ensembles that used in statistical thermodynamics calculation. In fact, these ensembles are just being implemented into the code of the simulations. With a suitable force field, a set of potential functions that describe the interactions between particles of the simulations, molecular simulations are useful to study biological systems. For example, the interaction between anesthetics and lipid bilayer is such a system, particularly, since the relative short reaction time period (~10 ns) involved in the interaction. However, the computational studies are capable to record trajectory in such time scale. However, it is worthy mentioned that drawback of MD simulation is the current computational power one can still only access to maximum to milliseconds time scale of simulations. With a time resolutions of a few femto-seconds, the access to ten to hundreds of nanosecond is suitable to study the biological system such as anesthetics binding to lipid bilayer. It is useful to understand the structural information and energetics of the interaction. Nevertheless, the computational investigations still allow us to investigate the mechanism of anesthetics biding to lipid bilayer membranes, in which is the example used in this article. Free energy is a very useful measurement of chemical reactions. The free energy difference between different states gives fundamental information of the thermodynamics. In addition to molecular dynamics simulations for studying certain chemical interactions, free energies calculations have been also well developed. Umbrella sampling, Thermodynamical Integration (TI) calculations, Steered Molecular Dynamics (SMD) (4) calculations, Adaptive Biasing Force (ABF) (5) calculations, and Alchemical Free Energy Perturbation (FEP) calculations are popular methods of free energy calculations. In this article, example will be the free energy calculation of transporting an anesthetics molecule through a biomembrane composed by lipid. In terms of studying the binding of anesthetics to lipid blilayer, free energy is one key component to reveal the energetics of the reaction. The free energy difference can be calculated for systems such anesthetics binding to lipid bilayer to study the energetics. In particular, the example free energy calculation is performed using Adaptive Biasing Force (ABF) method implemented in NAMD to determine the free energy profile, A(z), along the reaction coordinate (z), which is defined as the bilayer normal. Free energy, A, is important measurement in chemical interactions. It can be calculated along a reaction coordinate, z, as following (5): A = β-1ln Pz (#3) where Pz is the probability density of system of interest at z. Free energy differences, A(z) –A(0), can be obtained by integrate the derivative ∇A(z), the potential of mean force ∇A(z) = ∫ F(z) dz = - < Fz >z, (#4) where <Fz>z is called the mean force along the reaction coordinate (z).
Method and Case Study
NAMD is used for all the molecular dynamics (6) simulations reported here. Two systems were constructed: First system of 60 POPC lipids (30 per leaflet) and 3528 water molecules and it is used for validation purpose for the lipid membrane properties such as area per head-group, lipid thickness, and electron density profile. The second system was used to study drug-lipid interactions; see Figure 3. This system contains 60 POPC lipids, 3528 water molecules and 2 dibucaine molecules with chlorine counter ions, which are placed separately on each side of the lipid bilayer. The molecular dynamics simulations were performed using the general all-atom amber force field (GAFF) and RESP charges were calculated for dibucaine molecule and POPC (7,8,9). The TIP3P water model was used here to represent the water molecules. All simulations were performed using time step of 0.5 femtosecond. Pressure and temperature were controlled using Langevin dynamics fixed at 1 atm and 300 Kevin, respectively. A cut-off of 15Å is used for short range Lennard-Jones interactions. The particle-mesh Ewald method was employed for computing long-range electrostatic interactions. Equilibration was first started with NVT ensemble to remove bad contacts and then NPT ensemble with applied 80 dyn/cm surface tension (NPγT ensemble) was used for equilibration, this allowed for the area per head-group and lipid thickness to reach value that were comparable to the experimental values. 20 ns of NVT ensemble simulations were performed after equilibration for data analysis. Adaptive Biasing Force (ABF) method implemented in NAMD is used to determine the free energy profile, A(z), along the reaction coordinate (z), which is defined as the bilayer normal. Both free energy profiles of transporting one water molecule and one dibucaine molecule across the POPC lipid bilayer were reported here. Only a portion of reaction coordinate, from the bulky water region to the center of lipid core, was used for free energy calculation to reduce computational expense. The other half free energy profile was assumed to be mirror images to the first half. For the water transferring ABF calculations, the bilayer normal was divided into 6 even regions with the width of 5Å. For each window 5-7 ns ABF simulations were performed. In similar manner, for the dibucaine transferring system, the bilayer normal was divided into 10 even regions with the width of 3Å. For each window 7-10 ns ABF calculations were carried out. Simulations were extended in the region of near lipid membrane center for additional 5ns to obtain higher statistical accuracy. The area per head-group, lipid membrane thickness, electron density profile, and lipid order parameter profile were calculated based on the NPγT and NVT trajectories of either the lipid-only or drug-lipid systems as descried in the method section. The area per head-group and membrane thickness are used widely for validation and/or calibration for molecular dynamics simulations. For the lipid-only system, 64.2 ± 0.63 Å2is obtained and it is comparable with experimental values (68.3Å2). There was a small increment when dibucaine molecules were inserted into the lipid membrane (67.9 ± 0.92Å2was reported). Similarly, for the membrane thickness, 38.2 ± 0.36Å for lipid-only system and 36.8 ± 0.35Å for drug-lipid system were obtained using the average distance of the phosphorous atom to phosphorous atom from opposite sides of lipid bilayer. It also yields comparable results to the experimental value (37 Å).With less than 7% difference form experimental values, both the lipid-only system and drug-lipid system showed reasonable agreement in terms of area per head-group and lipid thickness. This confirmed that we could calculate the bulky properties of system using statistical thermodynamics by using computers to aid us with the heavy calculations. Electron density calculated from molecular dynamics simulations can be directly compared to small angle X-ray experimental data. The electron density profiles presented here was centered with respect to the membrane core (place center of mass of lipid membrane at zero) and were calculated using the last 20ns NVT trajectory. The electrons of hydrogen atoms were incorporated into heaver atoms to computer the electron profiles with bin size of 0.1 Å along the bilayer normal. Heavy atoms’ electrons are represented by added up electrons along the bilayer normal with every 0.1 Å bin size. For example, CH3 group counts 15 and NH group counts 16. Comparison of electron density profiles of experimental and MD simulations shows that, again, we could computational calculations to link experimental data with the calculation based on the statistical thermodynamics. The free energies of transferring one water molecule and one dibucaine molecule from water to the lipid bilayer core were calculated using ABF method implemented in NAMD, as descried above. Translocate one water molecule from water through lipid membrane was carried out as a reference point to compare to the free energy of transferring the dibucaine molecule through membrane. The reaction coordinate was set to zero at the core of lipid membrane and increase as the distant along the bilayer normal. Only half of the reaction coordinate was used for calculation and the other half was assumed to be mirror image respect to center of the membrane. It is found that to transfer one water molecule from water region to the center of the membrane core required 8.3 kcal/mol and it required 14.0 kcal/ mol to transfer one dibucaine molecule through the lipid bilayer. Both the profile and barrier height are in agreement to literature value (10). In this study, the ensemble used for free energy calculation is NVT ensemble. Therefore the outcome free energy estimated is the corresponding Helmholtz free energy. Once again, the statistical thermodynamics calculations are linked to experiments by computational calculations.
Conclusion
The computational study on particular example, dibucaine binding to lipid bilayer, listed in this article demonstrates the point of using statistical thermodynamics to calculate the bulky properties of the system. The MD simulations provide the evidences that the structural information of the system could be calculated for the molecular level using statistical thermodynamics calculation that is implemented in the computer codes. The results of MD simulations are all comparable to experimental value simply implies that the statistical thermodynamics calculations could predicted the bulky properties from a microscopic point of view. In addition, the free energies calculations also proved that the ensemble average and potential mean force calculation could be utilized to estimate the corresponding thermodynamic parameter. In particular, the ensemble for the free energy calculation is the NVT ensemble and the NVT ensemble calculation yields the Helmholtz free energy. To conclude, through statistical thermodynamics, we could calculate the bulky properties of system interested only using a few mathematical rules and some important ideas in statistical thermodynamic. With the computing powers, one can link the theoretical value obtain from statistical thermodynamic calculations to experimental values.
Bibliography
1) Engel, T & Reid, P Thermodynamics, Statistical Thermodynamis, & Kinetics 2005 2) McQuarrie, D Statistical Mechanics 1973 3) Alder, B. J.; T. E. Wainwright. Studies in Molecular Dynamics. I. General Method. J. Chem. Phys. 31 (2): 459. Bibcode1959 JChPh..31..459A. doi:10.1063/1.1730376 4) J. C. Phillips, R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, and K. Schulten 2005. Scalable moleculaar dynamics with NAMD. J. Comput. Chem. 26, 1781. 5) J. Henin, and C. Chipot 2004. Overcoming free energy barriers using unconstrained molecular dynamics simulaions J. Chem. Phys. 124:2904-2914 6) J. C. Phillips, R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, and K. Schulten 2005. Scalable moleculaar dynamics with NAMD. J. Comput. Chem. 26, 1781. 7) C. I. Bayly, P. Cieplak, W. D. Cornell, and P. A. Kollman 1993. A well-behaced electrostatic potential based method using charge restraints for deriving atomic charges: the RESP model. DOI: 10.1021/j100142a004. 8) W. D. Cornell, P. Cieplak, C. I. Bayly, and P. A. Kollman, 1993. Application of RESP charges to calculate conformational energies, hydrogen bond energies, and free energies of solvation. J. Am. Chem. Soc. 115:9620-9631 DOI:10.1021/ja00074a030 9) J. P. M. Jämbeck, F. Mocci, A P. Lyubartsev, A Laaksonen, 2012. Partial atomic charges and their impact on the free energy of solvation DOI: 10.1002/jcc.23117 10) K. Shinoda, W. Shinoda, and M. Mikami 2008. Efficient Free Energy Calculation of Water Across Lipid Membranes. Journal of Computational Chemistry DOI: 10.1002/jcc.20956 11) H.S. Hendrickson 1976. The penetration of local anesthetics into phosphatidylcholine monolayers Journal of Lipid Research 17: 393-398
12) H I Petrache, S W Dodd, and M F Brown, Area per lipid and acyl length distributions in fluid phosphatidylcholines determined by (2)H NMR spectroscopy. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1301193/pdf/11106622.pdf
13) Berger O, Edholm O, Jähnig F. Molecular dynamics simulations of a fluid bilayer of dipalmitoylphosphatidylcholine at full hydration, constant pressure, and constant temperature. Biophys J. 1997 May;72(5):2002–2013. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1184396/pdf/biophysj00034-0092.pdf 14) K.Y. Sanbonmatsu High Performance Computing in Biology, http://www.sciencedirect.com/science/article/pii/S104784770600308X
15) Echeverria I, Amzel LM, 2012 Estimation of free-energy differences from computed work distributions: an application of Jarzynski's equality DOI: 10.1021/jp300527q
16) Bradley M. Dickson, He Huang , and Carol Beth Post Unrestrained Computation of Free Energy along a Path, 2012 DOI: 10.1021/jp304720m
17) Levi C.T. Pierce , Romelia Salomon-Ferrer, Cesar Augusto F. de Oliveira, J. Andrew McCammon, and Ross C. Walker Routine Access to Millisecond Time Scale Events with Accelerated Molecular Dynamics, 2012. J. Chem. Theory Comput., 2012, 8 (9), pp 2997–3002 DOI: 10.1021/ct300284c
18) James C. Gumbart, Benot Roux, and Christophe Chipot Standard Binding Free Energies from Computer Simulations: What Is the Best Strategy? 2012, J. Chem. Theory Comput., Article ASAP DOI: 10.1021/ct3008099
19) Christophe Chipot Milestones in the Activation of a G Protein-Coupled Receptor. Insights from Molecular-Dynamics Simulations into the Human Cholecystokinin Receptor-1 2007, DOI: 10.1021/ct800313k
20) Jerome Henin, Giacomo Fiorin, Christophe Chipot and Michael L. Klein Exploring Multidimensional Free Energy Landscapes Using Time-Dependent Biases on Collective Variables, 2010, J. Chem. Theory Comput., 2010, 6 (1), pp 35–47 DOI: 10.1021/ct9004432
21) Andreas W. Gotz, Mark J. Williamson, Dong Xu, Duncan Poole, Scott Le Grand, and Ross C. Walker Routine Microsecond Molecular Dynamics Simulations with AMBER on GPUs. 1. Generalized Born, 2012 J. Chem. Theory Comput., 2012, 8 (5), pp 1542–1555 DOI: 10.1021/ct200909j
22) Bill R. Miller , III, T. Dwight McGee , Jr., Jason M. Swails, Nadine Homeyer, Holger Gohlke, and Adrian E. Roitberg MMPBSA.py: An Efficient Program for End-State Free Energy Calculations, J. Chem. Theory Comput., 2012, 8 (9), pp 3314–3321 DOI: 10.1021/ct300418h
Computational Calculations: a linkage of statistical thermodynamics and experiments
Introduction
Physical chemists usually use thermodynamic parameters to describe systems of interest. Typically temperature, pressure, and volume are used since these are the macroscopic variables that could be directly measured in the laboratories. Furthermore, only two out of the three variables mentioned above are needed to measure and thereby the studied systems could be described completely (1). Thermodynamics is indeed a very powerful tool for studying Chemistry under the macroscopic point of view. Not only provides direct quantities that could be measured from experiments, Thermodynamics could also show mathematical relationships between these quantities thereby we could use these relationship to understand the characteristics or properties of the systems under investigation. However, in the study of Thermodynamics, we do not consider the magnitude of the parameters and we also do not need to care about the fundamental particles which are the basic components of the system (2). From a molecular point of view, we need to describe the studied system in more details and we also need to consider the existence of atoms and molecules. Another area called statistical thermodynamics therefore is needed to bring into the study of Chemistry. Statistical thermodynamics is a discipline that has to deal with the calculation of bulky properties from a molecular point of view (2). In statistical thermodynamics, the concepts of ensembles, partition functions, Boltzmann statistics, and etc. are used to describe systems. We theoretical could calculate whatever system we would like to study. However, it is not practical to do so most of the time. First, assumptions still are needed to put into equations when the calculations are carried out in the study of statistical thermodynamics. Second, there are still some unsolved conceptual problem in statistical thermodynamics such as one is dealing with non-equilibrium systems although there have been so much effort were put into such kinds of system since mid-20th-century. Third, most importantly, it is impossible to carry out calculation on some complicated systems such as biological systems. Fortunately, since the World War II computers were introduced to scientist and thereby methods of calculations using statistical thermodynamics were started to developed. Computers are not necessary smarted than human but they are very fast of calculating equations and computers are good at not making mistakes. With the increasing computing power, scientists could study biological systems in more detail than ever before. For example, computational calculations of solving Newton's second law, Monte Carlo method, Potential Mean force calculations, and etc. are able to carry out to investigate complex system such protein folding and ion channel conformational change. Not being the calculations are impractical to perform by hand, it is also important to point out that it is typically a huge number of sample calculations we need to do and then take the average of these calculations, and out of such average we could minimize the statistical error or fluctuation in which we could be certain of our calculation. Such calculations are extremely long and are almost impossible to perform without computers. With the statistical thermodynamics in hand as a theoretical background and the utilization of advance computer developments, scientists therefore can study chemistry in the molecular level quantitatively. This article's focus is to introduce the methods of computational calculations and analysis of the results based of statistical thermodynamics calculations. The examples include molecular dynamics simulations, potential mean force calculations, and their role of being the linkage of experiments and statistical thermodynamics. More precisely, a project on computational investigation on a biological system will be given in this article as an example to show how theoretical calculations was done in the background of statistical thermodynamics and how the results are compared to the experimental works.
Background
In order to calculate the macroscopic properties from the microscopic point of view, some important mathematical concepts are needed to be introduced. First mathematics concept is the probability distribution. It is important to understand the random variables and the distribution function which are needed to be introduced in the calculations of statistical thermodynamics. The average, or the mean value, is a typical value obtained from calculation in the study of statistical thermodynamics. Moreover, the maximum term method is another concept will be applied when dealing with statistical thermodynamics. Since large numbers are usually involved in statistical thermodynamics calculation, it is appropriated to use the largest term, typically much larger comparing to other terms, as the peak out of all the probability values. The third mathematical concept is the Stirling's approximation because in statistical thermodynamics we usually have to deal with factorials of large numbers. The Stirling’s approximation (2) is the following,
lnN! = NlnN-N, where N is a very large number (#1)
Binomial and multinomial distributions, which are needed to use where the ensemble idea is used in statistical thermodynamics, could be used to determine how many states we encounter of many distinguishable system in a group under investigation. Especially the binomial distribution for large number is needed for statistical thermodynamics calculations because it is very important that one cal find the maximum value of functions. Finally, the method of Lagrange multipliers comes handy when we have to maximize variables since most of the times we need to find the maximum of certain functions or to set the derivative of certain function to be zero and solve for needed variables. With all the mathematical concepts above on mind, now the base of statistical thermodynamics calculation is set and we are ready to introduce some important concepts of statistical thermodynamics.
One of the important concepts that statistical thermodynamics introduced is ensemble average (2). To view the macroscopic properties such temperature, pressure, and volume from the molecular point of view, the idea of ensemble is introduced in the late 19th century. To calculate the bulky properties from the microscopic quantities, scientists introduced the concept of ensemble, in which the studied system is composed of a large numbers of identical subsystems. And these subsystems are then treated as equal parts of the studies systems. What we have to do is just to calculate the properties of one subsystem out of all these subsystems to understand the whole studied system. Popular ensembles (2) used in statistical thermodynamics are microcanonical ensemble, canonical ensemble, grandcanonical ensemble, and isothermal-isobaric. Microcanonical ensemble (NVE) is system consist constant number of particle (N) with fixed energy (E) and a fixed volume (V). Under the constrain of constant number of particle and fixed energy and fixed volume, the partition function with correlated thermodynamics parameter could be able to identified using the concept of ensemble average and mathematic concepts mentioned above. Similarly to microcanonical ensemble, grandcanonical (µVT) ensemble, canonical (NVT) ensemble, isothermal-isobaric (NPT) ensemble could be also used to calculate macroscopic properties, where µ is the chemical potential, T is the temperature, and P is the pressure (2), representatively.
The most important thing we could be able to get out of ensemble average is the partition function of a system because using the partition function one can derive the thermodynamic function and thereby calculate the thermodynamics properties. Therefore the bulky properties could be calculated from the point of view in the molecular level.
With the concepts of statistical thermodynamics and mathematics introduced, we have to realized that the ensemble average calculations is easy but time consuming when we have to deal with large and complicated system such as biological system. It is not practical to do such calculations over and over again by hand and it is almost impossible to do so most of the case. There is where computational calculation comes into place. The calculations performed on computers involves molecular dynamics (3) (MD) simulation on reaction dynamics, quantum mechanic calculations on electronic structural information, Monte Carlo calculation on random walk simulations, potential mean force calculation for estimating reaction free energies, and etc. In this article, MD simulation and PMF calculations will be described as the linkage of experiments and statistical thermodynamics.
Many experimental works had been carried out to investigate the mechanism of certain biological system such as anesthetics binding to lipid, however, due to the limitation of experiments, some perspectives in the molecular level are difficult to determine. For example, the binding mode of reaction is usually hard to reveal since the binding time is relative short and it is hard to detect by instruments. To access molecular detail, molecular dynamics simulations can be utilized. In molecular dynamics, the atoms interact with each other respect to time. The calculations of the motion of these atoms are solved using the Newton’s second law of motion:
F(x) = m ×a = -∇U(x) (#2)
where in eqution (#2), F(x) is function of the force act on particle respect to the position, x , m is the mass of particle, a is the acceleration of the particle and ∇U(x) is the potential energy of the particle.
These coupled differential equations are solved using numerical integration on the computers. From the initial position and velocity, MD simulations are used to calculate or predict the future positions, velocities, and potential energies. The time revolution of position and velocity is called the trajectory.
Typically, the simulations are using either one of the following ensembles: NVE, NVT, or NPT ensemble, whereas N, V, P, T,and E stands for constant numbers of atoms, volume, pressure, temperature, and energy, respectively. These ensembles are any different from the ensembles that used in statistical thermodynamics calculation. In fact, these ensembles are just being implemented into the code of the simulations. With a suitable force field, a set of potential functions that describe the interactions between particles of the simulations, molecular simulations are useful to study biological systems. For example, the interaction between anesthetics and lipid bilayer is such a system, particularly, since the relative short reaction time period (~10 ns) involved in the interaction. However, the computational studies are capable to record trajectory in such time scale. However, it is worthy mentioned that drawback of MD simulation is the current computational power one can still only access to maximum to milliseconds time scale of simulations. With a time resolutions of a few femto-seconds, the access to ten to hundreds of nanosecond is suitable to study the biological system such as anesthetics binding to lipid bilayer. It is useful to understand the structural information and energetics of the interaction. Nevertheless, the computational investigations still allow us to investigate the mechanism of anesthetics biding to lipid bilayer membranes, in which is the example used in this article.
Free energy is a very useful measurement of chemical reactions. The free energy difference between different states gives fundamental information of the thermodynamics. In addition to molecular dynamics simulations for studying certain chemical interactions, free energies calculations have been also well developed. Umbrella sampling, Thermodynamical Integration (TI) calculations, Steered Molecular Dynamics (SMD) (4) calculations, Adaptive Biasing Force (ABF) (5) calculations, and Alchemical Free Energy Perturbation (FEP) calculations are popular methods of free energy calculations. In this article, example will be the free energy calculation of transporting an anesthetics molecule through a biomembrane composed by lipid. In terms of studying the binding of anesthetics to lipid blilayer, free energy is one key component to reveal the energetics of the reaction. The free energy difference can be calculated for systems such anesthetics binding to lipid bilayer to study the energetics.
In particular, the example free energy calculation is performed using Adaptive Biasing Force (ABF) method implemented in NAMD to determine the free energy profile, A(z), along the reaction coordinate (z), which is defined as the bilayer normal.
Free energy, A, is important measurement in chemical interactions. It can be calculated along a reaction coordinate, z, as following (5):
A = β-1ln Pz (#3)
where Pz is the probability density of system of interest at z.
Free energy differences, A(z) –A(0), can be obtained by integrate the derivative ∇A(z), the potential of mean force
∇A(z) = ∫ F(z) dz = - < Fz >z, (#4)
where <Fz>z is called the mean force along the reaction coordinate (z).
Method and Case Study
NAMD is used for all the molecular dynamics (6) simulations reported here. Two systems were constructed: First system of 60 POPC lipids (30 per leaflet) and 3528 water molecules and it is used for validation purpose for the lipid membrane properties such as area per head-group, lipid thickness, and electron density profile. The second system was used to study drug-lipid interactions; see Figure 3. This system contains 60 POPC lipids, 3528 water molecules and 2 dibucaine molecules with chlorine counter ions, which are placed separately on each side of the lipid bilayer. The molecular dynamics simulations were performed using the general all-atom amber force field (GAFF) and RESP charges were calculated for dibucaine molecule and POPC (7,8,9). The TIP3P water model was used here to represent the water molecules. All simulations were performed using time step of 0.5 femtosecond. Pressure and temperature were controlled using Langevin dynamics fixed at 1 atm and 300 Kevin, respectively. A cut-off of 15Å is used for short range Lennard-Jones interactions. The particle-mesh Ewald method was employed for computing long-range electrostatic interactions. Equilibration was first started with NVT ensemble to remove bad contacts and then NPT ensemble with applied 80 dyn/cm surface tension (NPγT ensemble) was used for equilibration, this allowed for the area per head-group and lipid thickness to reach value that were comparable to the experimental values. 20 ns of NVT ensemble simulations were performed after equilibration for data analysis.
Adaptive Biasing Force (ABF) method implemented in NAMD is used to determine the free energy profile, A(z), along the reaction coordinate (z), which is defined as the bilayer normal. Both free energy profiles of transporting one water molecule and one dibucaine molecule across the POPC lipid bilayer were reported here.
Only a portion of reaction coordinate, from the bulky water region to the center of lipid core, was used for free energy calculation to reduce computational expense. The other half free energy profile was assumed to be mirror images to the first half. For the water transferring ABF calculations, the bilayer normal was divided into 6 even regions with the width of 5Å. For each window 5-7 ns ABF simulations were performed. In similar manner, for the dibucaine transferring system, the bilayer normal was divided into 10 even regions with the width of 3Å. For each window 7-10 ns ABF calculations were carried out. Simulations were extended in the region of near lipid membrane center for additional 5ns to obtain higher statistical accuracy.
The area per head-group, lipid membrane thickness, electron density profile, and lipid order parameter profile were calculated based on the NPγT and NVT trajectories of either the lipid-only or drug-lipid systems as descried in the method section. The area per head-group and membrane thickness are used widely for validation and/or calibration for molecular dynamics simulations. For the lipid-only system, 64.2 ± 0.63 Å2is obtained and it is comparable with experimental values (68.3Å2). There was a small increment when dibucaine molecules were inserted into the lipid membrane (67.9 ± 0.92Å2was reported). Similarly, for the membrane thickness, 38.2 ± 0.36Å for lipid-only system and 36.8 ± 0.35Å for drug-lipid system were obtained using the average distance of the phosphorous atom to phosphorous atom from opposite sides of lipid bilayer. It also yields comparable results to the experimental value (37 Å).With less than 7% difference form experimental values, both the lipid-only system and drug-lipid system showed reasonable agreement in terms of area per head-group and lipid thickness. This confirmed that we could calculate the bulky properties of system using statistical thermodynamics by using computers to aid us with the heavy calculations.
Electron density calculated from molecular dynamics simulations can be directly compared to small angle X-ray experimental data. The electron density profiles presented here was centered with respect to the membrane core (place center of mass of lipid membrane at zero) and were calculated using the last 20ns NVT trajectory. The electrons of hydrogen atoms were incorporated into heaver atoms to computer the electron profiles with bin size of 0.1 Å along the bilayer normal. Heavy atoms’ electrons are represented by added up electrons along the bilayer normal with every 0.1 Å bin size. For example, CH3 group counts 15 and NH group counts 16. Comparison of electron density profiles of experimental and MD simulations shows that, again, we could computational calculations to link experimental data with the calculation based on the statistical thermodynamics.
The free energies of transferring one water molecule and one dibucaine molecule from water to the lipid bilayer core were calculated using ABF method implemented in NAMD, as descried above. Translocate one water molecule from water through lipid membrane was carried out as a reference point to compare to the free energy of transferring the dibucaine molecule through membrane. The reaction coordinate was set to zero at the core of lipid membrane and increase as the distant along the bilayer normal. Only half of the reaction coordinate was used for calculation and the other half was assumed to be mirror image respect to center of the membrane. It is found that to transfer one water molecule from water region to the center of the membrane core required 8.3 kcal/mol and it required 14.0 kcal/ mol to transfer one dibucaine molecule through the lipid bilayer. Both the profile and barrier height are in agreement to literature value (10). In this study, the ensemble used for free energy calculation is NVT ensemble. Therefore the outcome free energy estimated is the corresponding Helmholtz free energy. Once again, the statistical thermodynamics calculations are linked to experiments by computational calculations.
Conclusion
The computational study on particular example, dibucaine binding to lipid bilayer, listed in this article demonstrates the point of using statistical thermodynamics to calculate the bulky properties of the system. The MD simulations provide the evidences that the structural information of the system could be calculated for the molecular level using statistical thermodynamics calculation that is implemented in the computer codes. The results of MD simulations are all comparable to experimental value simply implies that the statistical thermodynamics calculations could predicted the bulky properties from a microscopic point of view. In addition, the free energies calculations also proved that the ensemble average and potential mean force calculation could be utilized to estimate the corresponding thermodynamic parameter. In particular, the ensemble for the free energy calculation is the NVT ensemble and the NVT ensemble calculation yields the Helmholtz free energy. To conclude, through statistical thermodynamics, we could calculate the bulky properties of system interested only using a few mathematical rules and some important ideas in statistical thermodynamic. With the computing powers, one can link the theoretical value obtain from statistical thermodynamic calculations to experimental values.
Bibliography
1) Engel, T & Reid, P Thermodynamics, Statistical Thermodynamis, & Kinetics 2005
2) McQuarrie, D Statistical Mechanics 1973
3) Alder, B. J.; T. E. Wainwright. Studies in Molecular Dynamics. I. General Method. J. Chem. Phys. 31 (2): 459. Bibcode 1959 JChPh..31..459A. doi:10.1063/1.1730376
4) J. C. Phillips, R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, and K. Schulten 2005. Scalable moleculaar dynamics with NAMD. J. Comput. Chem. 26, 1781.
5) J. Henin, and C. Chipot 2004. Overcoming free energy barriers using unconstrained molecular dynamics simulaions J. Chem. Phys. 124:2904-2914
6) J. C. Phillips, R. Braun, W. Wang, J. Gumbart, E. Tajkhorshid, E. Villa, C. Chipot, R. D. Skeel, and K. Schulten 2005. Scalable moleculaar dynamics with NAMD. J. Comput. Chem. 26, 1781.
7) C. I. Bayly, P. Cieplak, W. D. Cornell, and P. A. Kollman 1993. A well-behaced electrostatic potential based method using charge restraints for deriving atomic charges: the RESP model. DOI: 10.1021/j100142a004.
8) W. D. Cornell, P. Cieplak, C. I. Bayly, and P. A. Kollman, 1993. Application of RESP charges to calculate conformational energies, hydrogen bond energies, and free energies of solvation. J. Am. Chem. Soc. 115:9620-9631 DOI:10.1021/ja00074a030
9) J. P. M. Jämbeck, F. Mocci, A P. Lyubartsev, A Laaksonen, 2012. Partial atomic charges and their impact on the free energy of solvation DOI: 10.1002/jcc.23117
10) K. Shinoda, W. Shinoda, and M. Mikami 2008. Efficient Free Energy Calculation of Water Across Lipid Membranes. Journal of Computational Chemistry DOI: 10.1002/jcc.20956
11) H.S. Hendrickson 1976. The penetration of local anesthetics into phosphatidylcholine monolayers Journal of Lipid Research 17: 393-398
12) H I Petrache, S W Dodd, and M F Brown, Area per lipid and acyl length distributions in fluid phosphatidylcholines determined by (2)H NMR spectroscopy. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1301193/pdf/11106622.pdf
13) Berger O, Edholm O, Jähnig F. Molecular dynamics simulations of a fluid bilayer of dipalmitoylphosphatidylcholine at full hydration, constant pressure, and constant temperature. Biophys J. 1997 May;72(5):2002–2013.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1184396/pdf/biophysj00034-0092.pdf
14) K.Y. Sanbonmatsu High Performance Computing in Biology,
http://www.sciencedirect.com/science/article/pii/S104784770600308X
15) Echeverria I, Amzel LM, 2012 Estimation of free-energy differences from computed work distributions: an application of Jarzynski's equality DOI: 10.1021/jp300527q
16) Bradley M. Dickson, He Huang , and Carol Beth Post Unrestrained Computation of Free Energy along a Path, 2012
DOI: 10.1021/jp304720m
17) Levi C.T. Pierce , Romelia Salomon-Ferrer, Cesar Augusto F. de Oliveira, J. Andrew McCammon, and Ross C. Walker Routine Access to Millisecond Time Scale Events with Accelerated Molecular Dynamics, 2012. J. Chem. Theory Comput., 2012, 8 (9), pp 2997–3002 DOI: 10.1021/ct300284c
18) James C. Gumbart, Benot Roux, and Christophe Chipot Standard Binding Free Energies from Computer Simulations: What Is the Best Strategy? 2012, J. Chem. Theory Comput., Article ASAP DOI: 10.1021/ct3008099
19) Christophe Chipot Milestones in the Activation of a G Protein-Coupled Receptor. Insights from Molecular-Dynamics Simulations into the Human Cholecystokinin Receptor-1 2007, DOI: 10.1021/ct800313k
20) Jerome Henin, Giacomo Fiorin, Christophe Chipot and Michael L. Klein Exploring Multidimensional Free Energy Landscapes Using Time-Dependent Biases on Collective Variables, 2010, J. Chem. Theory Comput., 2010, 6 (1), pp 35–47
DOI: 10.1021/ct9004432
21) Andreas W. Gotz, Mark J. Williamson, Dong Xu, Duncan Poole, Scott Le Grand, and Ross C. Walker Routine Microsecond Molecular Dynamics Simulations with AMBER on GPUs. 1. Generalized Born, 2012 J. Chem. Theory Comput., 2012, 8 (5), pp 1542–1555 DOI: 10.1021/ct200909j
22) Bill R. Miller , III, T. Dwight McGee , Jr., Jason M. Swails, Nadine Homeyer, Holger Gohlke, and Adrian E. Roitberg MMPBSA.py: An Efficient Program for End-State Free Energy Calculations, J. Chem. Theory Comput., 2012, 8 (9), pp 3314–3321 DOI: 10.1021/ct300418h